# Velocity based frictional force equations

1. Sep 29, 2004

### abertram28

Im doing a problem with variable frictional forces.

My main equation is -mkv^2=F . We are to assume the force driving the object remains constant, kinda like a boat on the lake full bore.

So, I set my F=ma equation up.
-mkv^2=m(dv/dt)

Next I removed m and inverted both equations to solve for dt.
-dv/(kv^2)=dt

Next I intetegrated both sides seperately. I was taught to use a "dummy variable" by marking v and t somehow. I simply chose to use a superscript prime marking on my paper. anyhow... Ill use a little v for real velocity and big V for dummy velocity.
(1/kV)|0 to v = t

Isnt that (1/kv) - (1/0) ?

This equation doesnt solve nicely. In my setup I am given the equation for velocity and only asked to show how I got it.
V=Vo / (1 + Vo*kt)

Please help... I posted part of this problem over in classical when I had a different problem with it, so please dont flame me for double posting or spamming the board. If thats your opinion I couldnt care less.

TIA to anyone who helps!

2. Sep 29, 2004

### Pyrrhus

Let me see

$$F = -mkv^2$$

$$m \frac{dv}{dt} = -mkv^2$$

$$-\frac{dv}{kv^2} = dt$$

$$\int^{v}_{v_{o}} -\frac{dv}{kv^2} = \int^{t}_{0} dt$$

$$\frac{1}{kv}]^{v}_{v_{o}} = t]^{t}_{0}$$

$$\frac{1}{kv} - \frac{1}{kv_{o}}= t - 0$$

3. Sep 29, 2004

### abertram28

Hey, cyclovenom!

Thanks, all the examples we did in class used velocity starting at 0.. I didnt understand the part where we get limits of integration from. now it makes perfect sense, v=0 at t=0, so the lower limits are 0 and 0. in this case, v=Vo at t=0

Thanks for helping me out! Im totally clear, AND im going to start using latex!! woot!